is a very delicate matter. In our case, the analogue of \({\mathbf {x}}_{\tau }\)  is \((I-\eta H)^{t-1-\tau }(Z_{\tau } Z_{\tau } - I)(\nabla f(x_{\tau }) - \nabla f(x_{\tau }^{\prime }))\) , while the analogue of
\(\sigma _{\tau }^2\)  is \((1 + \eta \gamma )^{2(t- 1 -\tau )} \mathbb {E}\left[*{\left(Z_{\tau }Z_{\tau } - I\right) (\nabla f(x_{\tau }) - \nabla f(x_{\tau }^{\prime }))}^2 \mid {\mathcal {F}}_{\tau -1} \right]\) . Existing techniques (cf. [1]}, [2]}) rely crucially on subGaussian properties that allow for each \(\tau \)  the moment-generating function \(\mathbb {E}[e^{\theta {\mathbf {Y}}_{\tau }} \mid {\mathcal {F}}_{\tau - 1}]\)  to be defined for any fixed (and non-random) \(\theta > 0\) , where \({\mathbf {Y}}_{\tau }\)  takes the form
\( {\mathbf {Y}}_{\tau } = \begin{bmatrix}0 & {\mathbf {x}}_{\tau }^\top \\{\mathbf {x}}_{\tau } & 0\end{bmatrix},\) 
